Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392277 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14681 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,7)(4,16)(5,11)(6,10)(8,17)(9,14,24,19)(13,21,27,26)(18,30)(20,25)(22,32,29,28)(23,31);(1,3)(2,7)(4,13)(5,14)(6,15)(8,21)(9,11)(10,12)(16,26)(17,27)(18,28)(19,20)(22,23)(24,25)(29,30)(31,32);(1,4)(2,8,10,16)(3,13)(5,18)(6,17)(7,21,12,26)(9,22,24,29)(11,23,25,30)(14,28)(15,27)(19,32)(20,31);(1,3)(2,9)(4,13)(5,19)(6,15)(7,11)(8,22)(10,24)(12,25)(14,20)(16,29)(17,27)(18,32)(21,23)(26,30)(28,31)Polycyclic presentation:
2^9|a^2=h,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233776, #72319119, … |
| maximal normal subgroup of | #72233772, #72233776, #72319119, … |
| maximal quotient of | #72317846, #72317877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299473);